Finite Group Databasebeta

← #14536196

Subgroups — #14536196

Order 2176.

424 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11762750C211yesmaximal, characteristic, complemented
2#11762747C211yesmaximal, characteristic, complemented
3#11762729C211yesmaximal, characteristic, complemented
4#11762729C211yesmaximal, characteristic, complemented
5#11762721C211yesmaximal, characteristic
6#11762721C211yesmaximal, characteristic
7#11762714C211yesmaximal, characteristic, complemented
8#11762713C211yesmaximal, characteristic, complemented
9#11762703C211yesmaximal, characteristic
10#11762697C211yesmaximal, characteristic
11#11762682C211yesmaximal, characteristic
12#11762682C211yesmaximal, characteristic
13#11762673C211yesmaximal, characteristic, complemented
14#11762673C211yesmaximal, characteristic, complemented
15C17×#385C211yesmaximal, characteristic
16C23×C4×C17C2211yescharacteristic
17C22×Dic34C2211yescharacteristic
18C22×Dic34C2211yescharacteristic
19C2×C4⋊C4×C17C2211yescharacteristic
20C2×C4⋊C4×C17C2211yescharacteristic
21C2×C22⋊C4×C17C2211yescharacteristic
22C2×C22⋊C4×C17C2211yescharacteristic
23C2×C22⋊Dic17C2211yescharacteristic
24C2×C22⋊Dic17C2211yescharacteristic
25C2×C22⋊Dic17C2211yescharacteristic
26C2×C22⋊Dic17C2211yescharacteristic
27C2×#63225C2211yescharacteristic
28C2×#63225C2211yescharacteristic
29C2×#63225C2211yescharacteristic
30C2×#63225C2211yescharacteristic
31C2×#63225C2211yescharacteristic
32C2×#63225C2211yescharacteristic
33C2×C4×Dic17C2211yescharacteristic
34C2×C4×Dic17C2211yescharacteristic
35C17×#95C2211yescharacteristic
36C17×#95C2211yescharacteristic
37#10587502C2211yescharacteristic
38#10587502C2211yescharacteristic
39#10587502C2211yescharacteristic
40#10587502C2211yescharacteristic
41#10587502C2211yescharacteristic
42#10587502C2211yescharacteristic
43#10587502C2211yescharacteristic
44#10587502C2211yescharacteristic
45#10587502C2211yescharacteristic
46#10587502C2211yescharacteristic
47#10587502C2211yescharacteristic
48#10587502C2211yescharacteristic
49#10587502C2211yescharacteristic
50#10587502C2211yescharacteristic
51C24×C17C2311yescharacteristic
52C22×C4×C17—41no—
53C22×C4×C17—41no—
54C22×C4×C17—41no—
55C22×C4×C17C2311yescharacteristic
56C22×C4×C17C2311yescharacteristic
57C22×C4×C17C2311yescharacteristic
58C22×C4×C17C2311yescharacteristic
59C22×C4×C17C2311yescharacteristic
60C22×C4×C17C2311yescharacteristic
61C22×Dic17C2311yescharacteristic
62C22×Dic17C2311yescharacteristic
63C22×Dic17C2311yescharacteristic
64C22×Dic17C2311yescharacteristic
65C22×Dic17C2311yescharacteristic
66C22×Dic17C2311yescharacteristic
67C22×Dic17C2311yescharacteristic
68C22×Dic17C2311yescharacteristic
69C2×Dic34—22no—
70C2×Dic34—22no—
71C2×Dic34—41no—
72C2×Dic34—41no—
73C2×Dic34—41no—
74C2×Dic34—41no—
75C4⋊C4×C17—41no—
76C4⋊C4×C17—41no—
77C22⋊C4×C17—22no—
78C22⋊C4×C17—41no—
79C22⋊Dic17—22no—
80C22⋊Dic17—41no—
81C22⋊Dic17—41no—
82C22⋊Dic17Q814yescomplemented
83#63225—41no—
84#63225—41no—
85#63225—41no—
86#63225—41no—
87#63225—41no—
88#63225—41no—
89C4×Dic17—22no—
90C4×Dic17—22no—
91C23×C17—21no—
92C23×C17—21no—
93C23×C17—21no—
94C23×C17—21no—
95C23×C17—21no—
96C23×C17—21no—
97C23×C17C2×Q812yes—
98C23×C17C2411yescharacteristic
99C2×C4×C17—21no—
100C2×C4×C17—21no—
101C2×C4×C17—21no—
102C2×C4×C17—21no—
103C2×C4×C17—21no—
104C2×C4×C17—21no—
105C2×C4×C17—21no—
106C2×C4×C17—21no—
107C2×C4×C17—21no—
108C2×C4×C17—21no—
109C2×C4×C17—21no—
110C2×C4×C17—21no—
111C2×C4×C17—21no—
112C2×C4×C17—21no—
113C2×C4×C17—21no—
114C2×C4×C17—21no—
115C2×C4×C17—21no—
116C2×C4×C17—44no—
117C2×C4×C17#4112yes—
118C2×Dic17—21no—
119C2×Dic17—21no—
120C2×Dic17—21no—
121C2×Dic17—21no—
122C2×Dic17—21no—
123C2×Dic17—21no—
124C2×Dic17—21no—
125C2×Dic17—21no—
126C2×Dic17—21no—
127C2×Dic17—21no—
128C2×Dic17—21no—
129C2×Dic17—21no—
130C2×Dic17—21no—
131C2×Dic17—21no—
132C2×Dic17—21no—
133C2×Dic17—21no—
134C2×Dic17—21no—
135C2×Dic17—21no—
136C2×Dic17—21no—
137C2×Dic17—21no—
138C2×Dic17—21no—
139C2×Dic17C2×Q812yes—
140C2×Dic17C2×Q812yes—
141C2×Dic17#4112yes—
142Dic34—44no—
143Dic34—44no—
144#2763—171nomaximal, Sylow
145C22×C17—22no—
146C22×C17—22no—
147C22×C17—22no—
148C22×C17—41no—
149C22×C17—41no—
150C22×C17—41no—
151C22×C17—41no—
152C22×C17C22×Q811yescharacteristic
153C22×C17C2×#4111yescharacteristic
154C22×C17C2×#4111yescharacteristic
155C22×C17H(5,𝔽2)11yescharacteristic
156C22×C17H(5,𝔽2)11yescharacteristic
157C22×C17H(5,𝔽2)11yescharacteristic
158C22×C172−1+411yescharacteristic
159C4×C17—22no—
160C4×C17—41no—
161C4×C17—41no—
162C4×C17—41no—
163C4×C17—41no—
164C4×C17—41no—
165Dic17—22no—
166Dic17—22no—
167Dic17—22no—
168Dic17—41no—
169Dic17—41no—
170Dic17—41no—
171Dic17—41no—
172Dic17—41no—
173#400—171no—
174#400—171no—
175#395—171no—
176#395—171no—
177#393—171no—
178#393—171no—
179#391—171no—
180#391—171no—
181#388—171no—
182#388—171no—
183#385—171no—
184#385—171no—
185#385D1711yescharacteristic
186#382—171no—
187#382—171no—
188C2×C17—42no—
189C2×C17#54211yescharacteristic
190C2×C17#54211yescharacteristic
191C2×C17#54411yescharacteristic
192C2×C17#55111yescharacteristic
193C2×C17#55411yescharacteristic
194C2×C17#55411yescharacteristic
195C2×C17H(3,𝔽4)11yescharacteristic
196C22×Q8—171no—
197C22×Q8—171no—
198C23×C4C2×D1711yescharacteristic
199C2×C4⋊C4—171no—
200C2×C4⋊C4—171no—
201C2×C4⋊C4—171no—
202C2×C4⋊C4—171no—
203C2×C4⋊C4—171no—
204C2×C4⋊C4—171no—
205C2×C4⋊C4C2×D1711yescharacteristic
206C2×C4⋊C4C2×D1711yescharacteristic
207C2×C22⋊C4—171no—
208C2×C22⋊C4—171no—
209C2×C22⋊C4—171no—
210C2×C22⋊C4—171no—
211C2×C22⋊C4C2×D1711yescharacteristic
212C2×C22⋊C4C2×D1711yescharacteristic
213C2×C42—171no—
214C2×C42—171no—
215#95—171no—
216#95—171no—
217#95—171no—
218#95—171no—
219#95—171no—
220#95—171no—
221#95—171no—
222#95—171no—
223#95—171no—
224#95—171no—
225#95—171no—
226#95—171no—
227#95—171no—
228#95—171no—
229#95C2×D1711yescharacteristic
230#95C2×D1711yescharacteristic
231C17#276311yescharacteristic, complemented, Sylow
232C24C22×D1711yescharacteristic
233C2×Q8—342no—
234C2×Q8—342no—
235C2×Q8—681no—
236C2×Q8—681no—
237C2×Q8—681no—
238C2×Q8—681no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—171no—
243C22×C4—171no—
244C22×C4—171no—
245C22×C4—171no—
246C22×C4—171no—
247C22×C4—171no—
248C22×C4—171no—
249C22×C4—171no—
250C22×C4C22×D1711yescharacteristic
251C22×C4C22×D1711yescharacteristic
252C22×C4C22×D1711yescharacteristic
253C22×C4C22×D1711yescharacteristic
254C22×C4C22×D1711yescharacteristic
255C22×C4C22×D1711yescharacteristic
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—681no—
259C4⋊C4—681no—
260C4⋊C4—681no—
261C4⋊C4—681no—
262C4⋊C4—681no—
263C4⋊C4—681no—
264C22⋊C4—22no—
265C22⋊C4—41no—
266C22⋊C4—174no—
267C22⋊C4—342no—
268C22⋊C4—681no—
269C22⋊C4—681no—
270C42—342no—
271C42—342no—
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23Q8×D1712yes—
279C23C23×D1711yescharacteristic
280Q8—684no—
281Q8—684no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—44no—
300C2×C4—172no—
301C2×C4—172no—
302C2×C4—172no—
303C2×C4—341no—
304C2×C4—341no—
305C2×C4—341no—
306C2×C4—341no—
307C2×C4—341no—
308C2×C4—341no—
309C2×C4—341no—
310C2×C4—341no—
311C2×C4—341no—
312C2×C4—341no—
313C2×C4—341no—
314C2×C4—341no—
315C2×C4—341no—
316C2×C4—341no—
317C2×C4—341no—
318C2×C4—341no—
319C2×C4—341no—
320C2×C4—341no—
321C2×C4—341no—
322C2×C4—341no—
323C2×C4—341no—
324C2×C4#6325212yes—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C2×#6325211yescharacteristic
333C22#1058769111yescharacteristic
334C22#1058769111yescharacteristic
335C22#1058769111yescharacteristic
336C22C2×Q8×D1711yescharacteristic
337C22#41×D1711yescharacteristic
338C22#1058769711yescharacteristic
339C4—22no—
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—342no—
346C4—342no—
347C4—342no—
348C4—681no—
349C4—681no—
350C4—681no—
351C4—681no—
352C4—681no—
353C2—42no—
354C2#1176342411yescharacteristic
355C2#1176342711yescharacteristic
356C2#1176343811yescharacteristic
357C2#1176348811yescharacteristic
358C2#1176351311yescharacteristic
359C2#1176351411yescharacteristic
360C2#1176355211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D17
commutator, socle, cosocleC23×C17C24
FittingC17×#385C2
prime radicalC24×C17C23
hypercenter#385D17
hypocenterC17#2763
LCSG ▷ C23×C17 ▷ C17
UCSC1 ◁ C23 ◁ #385

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #11762750C2semidirect
2. #11762747C2semidirect
3. #11762729C2semidirect
4. #11762729C2semidirect
7. #11762714C2semidirect
8. #11762713C2semidirect
13. #11762673C2semidirect
14. #11762673C2semidirect
82. C22⋊Dic17Q8semidirect
231. C17#2763semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#117627502
#117627472
#117627292
#117627212
#117627142
#117627132
#117627032
#117626972
#117626822
#117626732
C17×#3852
#276317